Particular solutions of 3 linear PDEs with nonstandart right-hand side

Dear **MHF **members,

I have the following 3 PDEs, whose solutions I know but I am looking for nice explanary solutions. Please help me.

- [Solution: ]
- [Solution: ]
- [Solution: ]

Thanks.

**bkarpuz**

Re: Particular solutions of 3 linear PDEs with nonstandart right-hand side

What do you mean by "nice explanary solutions"? Do you mean how they came up with ? If so, I think I can explain this.

Re: Particular solutions of 3 linear PDEs with nonstandart right-hand side

Quote:

Originally Posted by

**Danny** What do you mean by "nice explanary solutions"? Do you mean how they came up with

? If so, I think I can explain this.

Exactly!

Re: Particular solutions of 3 linear PDEs with nonstandart right-hand side

The second and third are rather straight forward. For the second, the RHS is only a function of y so sek a particular solution in the form You'll get an ODE for (and it's linear!) Similarly for the third. For the first, you'll notice that the RHS is a function of so seek Since the LHS of the PDEs are independent of both independent variables, this method works.

Hope it helps.

Re: Particular solutions of 3 linear PDEs with nonstandart right-hand side

Quote:

Originally Posted by

**Danny** The second and third are rather straight forward. For the second, the RHS is only a function of y so sek a particular solution in the form

You'll get an ODE for

(and it's linear!) Similarly for the third. For the first, you'll notice that the RHS is a function of

so seek

Since the LHS of the PDEs are independent of both independent variables, this method works.

Hope it helps.

For the first one, I also noticed that as the left-hand side is involves only 2nd order derivatives.

Dor the 2nd and the 3rd ones, I rewrote left-hand side as and or but I could not see any nice things here...

Is it possible to make change of variables to reduce the equation into first-order equation as in the ordinary case?

Thanks...

Re: Particular solutions of 3 linear PDEs with nonstandart right-hand side

As you've said, for the first, you could made a change of variables where u = 2x + y and pick another independent variable.

For the second and third, since you've noticed that the operators factor, you can introduced a new variable as to make your problem two linear first order PDEs. For example, for the second one, if you let

then

Solve the second, then use this in the first.